On The Communication Complexity of Finding an (Approximate) Stable Marriage

TL;DR

This paper proves that finding a stable or approximately stable matching in the stable marriage problem requires quadratic communication, establishing the optimality of Gale-Shapley's algorithm and extending the bound to approximate solutions.

Contribution

It establishes a tight communication complexity lower bound for stable marriage protocols and introduces a metric for approximate stability with corresponding bounds.

Findings

Any protocol for stable marriage requires Ω(n^2) bits of communication.

Gale-Shapley's algorithm is communication-optimal up to a logarithmic factor.

Approximate stability also requires Ω(n^2) communication, matching the exact case.

Abstract

In this paper, we consider the communication complexity of protocols that compute stable matchings. We work within the context of Gale and Shapley's original stable marriage problem\cite{GS62}: $n$ men and $n$ women each privately hold a total and strict ordering on all of the members of the opposite gender. They wish to collaborate in order to find a stable matching---a pairing of the men and women such that no unmatched pair mutually prefer each other to their assigned partners in the matching. We show that any communication protocol (deterministic, nondeterministic, or randomized) that correctly ouputs a stable matching requires $\Omega(n^2)$ bits of communication. Thus, the original algorithm of Gale and Shapley is communication-optimal up to a logarithmic factor. We then introduce a "divorce metric" on the set of all matchings, which allows us to consider approximately stable matchings. We describe an efficient algorithm to compute the "distance to stability" of a given matching. We then show that even under the relaxed requirement that a protocol only yield an approximate stable matching, the $\Omega(n^2)$ communication lower bound still holds.